Bifurcation of critical points along gap-continuous families of subspaces

We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of critical points along these families, and apply our results to semilinear systems of ordinary differential equations

The Role Of Exotic Affine Spaces In the Classification Of Homogeneous Affine Varieties

Let G be a connected linear algebraic group over C and let H a closed algebraic subgroup. A fundamental problem in the study of homogeneous spaces is to describe, characterize, or classify those quotients G/H that are affine varieties. While cohomological characterizations of affine G/H are possible, there is still no general group-theoretic conditions that imply G/H is affine. In this article, we survey some of the known results about this problem and suggest a way of classifying affine G/H by means of its internal geometric structure as a fiber bundle. Cohomological characterizations of affine G/H provide useful vanishing theorems and related information if one already knows G/H is affine. Such characterizations cannot be realistically applied to prove that a given homogeneous space G/H is affine. Ideally, one would like to have easily verified group-theoretic conditions on G and H that imply G/H is affine. Very few positive results are known in this direction, the most notable of which is Matsushima’s Theorem for reductive groups. For general linear algebraic group